Sequences

1. Sequences 2, 4, 6, 8, 10, ….

2. Vocabulary • A Sequence is a function whose domain is the set of positive integers. {1, 1, 2, 3, 5, 8, …} • A Function is a relation where for each input there is exactly one output. • Domain is the set of all inputs in a function. • Positive integers : {1, 2, 3, 4, 5, …} • An explicit definition of a sequence tells us what the nth term of a sequence will be, given n. • i.e: an = 2n+1 • A recursive definition of a sequence tells us the next term of a sequence given the term before it. You always have to indicate the first term of the sequence. • i.e.: a1 = 3 an = an-1 +2

3. Arithmetic Sequences • When the difference between successive terms of a sequence is always the same number, the sequence is called arithmetic. • The recursive formula for an arithmetic sequence is: • a1 = a, an= an-1 + d, where d is the common difference, and a is the first term in the sequence. • The explicit formula for an arithmetic sequence is: • an = d(n-1)+a1 , where d is the common difference and a1 is the first term in the sequence.

4. Example • Determine if the following sequence is arithmetic, if it is find the common difference and write a recursive formula and explicit formula for the sequence. • {2, 4, 6, …} • {3, 6, 9, …}

5. Geometric Sequences • When the ratio (divide) of successive terms of a sequence is always the same nonzero number, the sequence is called geometric. • The recursive formula for a geometric sequence is: • a1 = a, an= ran-1, where r is the common ratio • The explicit formula for a geometric sequence is: • an = a1rn-1, where r is the common ratio and a1 is the first term in the sequence.